Chi-squared

Here again the quantity is the (1 — /3) percentile of a chi square distribution with V degrees of freedom. If only a 100(1 — a)% lower confidence limit is desired, it can be calculated from... 

Chi-Square Distribution For some industrial applications, produrt uniformity is of primary importance. The sample standard deviation. s is most often used to characterize uniformity. In dealing with this problem, the chi-square distribution can be used where = (.s /G ) (df). The chi-square distribution is a family of distributions which are defined by the degrees of freedom associated with the sample variance. For most applications, df is equal to the sample size minus 1. 

In terms of the tensile-strength table previously given, the respective chi-square sample values for the daily, weekly, and monthly figures couldbe computed. The corresponding df woiJdbe 4, 24, and 99 respec tively. These numbers would represent sample values from the respec tive distributions which are summarized in Table 3-6. 

The basic underlying assumption for the mathematical derivation of chi square is that a random sample was selected from a normal distribution with variance G. When the population is not normal but skewed, square probabihties could be substantially in error. 

The chi-square distribution can be applied to other types of apph-catlon which are of an entirely different nature. These include apph-cations which are discussed under Goodness-of-Fit Test and Two-Way Test for Independence of Count Data. In these applications, the mathematical formulation and context are entirely different, but they do result in the same table of values. 

The F distribution, similar to the chi square, is sensitive to the basic assumption that sample values were selected randomly from a normal distribution. 

Confidence Interval for a Variance The chi-square distribution can be used to derive a confidence interval for a population variance <7 when the parent population is normally distributed. For a 100(1 — Ot) percent confidence intei val... 

Crowe, C.M., Recursive Identification of Gross Errors in Linear Data Reconciliation, AJChE Journal, 34(4), 1988,541-550. (Global chi square test, measurement test)... 

The derivation will not be provided. Suffice it to say that the failures in a time interval may be modeled using the binomial distribution. As these intervals are reduced in size, this goes over to the Poisson distribution and the MTTF is chi-square distributed according to equation 2.9-31, where = 2 A N T and the degrees of freedom,/= 2(M+i). 

Confidence is calculated as the partial integral over the chi-squared distribution, i.e., the partial integral over equation 2.5-31 which is equation 2.5-32. where is the cumulative... 

Table 2.5-1 Values of the Inverse Cumulative Chi-Squared Distribution in Irrms of percentage confidence with M fuitun ...

It is important to note that the chi-squared estimator provides upper bounds on A for the case of zero failures. For example, a certain type of nuclear plant may have 115 plant-years of experience using 61 control rods. If there has never been a failure of a control rod, what is A for 50% (median) and 90% confidence ... 

Chapter 12 discusses the software provided with this book including the program Lambda that does the chi-squared and F-number calculations. 

This says that the failure rate is less than or equal to the inverse cumulative chi-squared distribution with confidence a and degrees of freedom equal to twice the number of failures including pseudo- failures divided by twice the time including psuedo-time. 

It may be decided that the gamma prior cannot be greater than a certain value xf. This has the effect of true Ling the normalizing denominator in equation 2.6-10," and leads to equation 2.6-17, where P(x v) is the cumulative integral from 0 to over the chi-squared density function with V degrees of freedom, a is the prescribed confidence fraction, and = 2 A" (t+Tr). Thus, the effect of the truncated gamma prior is to modify the confidence interval to become an effective confidence interval of a ... 

The numerator is a random normally distributed variable whose precision may be estimated as V(N) the percent of its error is f (N)/N = f (N). For example, if a certain type of component has had 100 failures, there is a 10% error in the estimated failure rate if there is no uncertainty in the denominator. Estimating the error bounds by this method has two weaknesses 1) the approximate mathematics, and the case of no failures, for which the estimated probability is zero which is absurd. A better way is to use the chi-squared estimator (equation 2,5.3.1) for failure per time or the F-number estimator (equation 2.5.3.2) for failure per demand. (See Lambda Chapter 12 ),... 

Uncertainly estimates are made for the total CDF by assigning probability distributions to basic events and propagating the distributions through a simplified model. Uncertainties are assumed to be either log-normal or "maximum entropy" distributions. Chi-squared confidence interval tests are used at 50% and 95% of these distributions. The simplified CDF model includes the dominant cutsets from all five contributing classes of accidents, and is within 97% of the CDF calculated with the full Level 1 model. 

Selection 2 is a similar calculation using the F-Number method (Section 2.5.3.2) 3 calculates the integral over the Chi-Squared distribution. When selected i nput the upper limit of integration... 

In order to determine the quality (or the validity) of fit of a particular function to the data points given, a comparison of the deviation of the curve from the data to the size of the experimental error can be made. The deviations (i.e., the scatter off the curve) should be of the same order of magnitude as the experimental error, so that the quantity chi-squared is defined as... 

The chi-square distribution gives the probability for a continuous random variable bounded on the left tail. The probability function has a shape parameter... 

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